Linear toral endomorphisms, exactness, Bernoulliness and generators

Abstract

Let $T$ be a non-invertible surjective continuous endomorphism of the group $\ttf^d\,:= \rrf^d/\zzf^d$. Then $T$ preserves the Haar measure, is associated to some $d \times d$ matrix $A$ with integer coefficients and is $r$-to-one, where $r = |\det A| \ge 2$. In~\cite{Krzyzewski}, Krzy\.{z}ewski gives a necessary and sufficient condition for $T$ to be exact, and shows that $T$ -- viewed as a measure-preserving map -- is isomorphic to the direct product of some (possibly trivial) toral automorphism by some exact toral endomorphism. We re-prove these results by more elementary and constructive methods, whereas Krzy\.{z}ewski actually works with endomorphisms of compact Abelian groups and relies on non-constructive theorems. Krzy\.{z}ewski's exactness condition and Mihailescu's papers ~\cite{Mihailescu 2012} and~\cite{Mihailescu 2013} show that $T$ is isomorphic to the one-sided uniform Bernoulli shift on $\odc 0,r-1 \fdc^{\zzf\_+}$ if and only if $A$ is expanding (i.e. the modulus of each eigenvalue is $>1$). When $A$ is not expanding and hyperbolic (i.e. all eigenvalues have modulus different from $1$), Mihailescu~\cite{Mihailescu 2012} shows that $T$ cannot have a generating Rokhlin partition. More generally, when $A$ is not expanding (whether hyperbolic or not), we show that the endomorphism $T$ cannot have a smooth finite generator (whether independent or not).